Fair Distribution of Sweets: Analysis & Comparison

Learning Outcomes
Learning Outcomes
Understanding the 
Understanding the 
Mean
Mean
 of a
 of a
 Distribution
 Distribution
through Fair Share.
through Fair Share.
Engaging with 
Engaging with 
Variability
Variability
 in a Distribution.
 in a Distribution.
Measuring Variability through counting the
Measuring Variability through counting the
amount of moves needed to make a
amount of moves needed to make a
Distribution fair.
Distribution fair.
Introducing 
Introducing 
Standard Deviation 
Standard Deviation 
as a more
as a more
sophisticated way of measuring Variability.
sophisticated way of measuring Variability.
Key Words
Key Words
Distribution
Distribution
Fair
Fair
Unfair
Unfair
Mean
Mean
Variability
Variability
Spread of a Distribution
Spread of a Distribution
The following represents a distribution of 45
The following represents a distribution of 45
sweets shared among 9 students.
sweets shared among 9 students.
Is this a fair distribution of the sweets?
Is this a fair distribution of the sweets?
Lets make it fair.
Lets make it fair.
We need to move around some sweets.
We need to move around some sweets.
How many times will we need to move a sweet
How many times will we need to move a sweet
 to make it fair?
 to make it fair?
 
3 moves
 
1 move
 
2 moves
How many moves?
How many moves?
 
6 moves
6 moves
 
How many sweets are in a Fair Share?
How many sweets are in a Fair Share?
 
5 sweets
5 sweets
 
We say “The Mean of the distribution is 5”
We say “The Mean of the distribution is 5”
What’s the Median of the Distribution?
What’s the Median of the Distribution?
What’s the Median of the Distribution?
What’s the Median of the Distribution?
 
5
5
Recap:  In the below Distribution of Sweets
Recap:  In the below Distribution of Sweets
A Fair Share/Mean = 5
A Fair Share/Mean = 5
No of Moves to make it fair = 6
No of Moves to make it fair = 6
Median = 5
Median = 5
Here are 6 more Distributions of the 45 sweets
Here are 6 more Distributions of the 45 sweets
Each row totals 45
Which one looks like the most 
Which one looks like the most 
fair
fair
 distribution?
 distribution?
Each row totals 45
Mean/Fair Share = 5
Moves to make fair = 2
Median = 5
Which one looks like the most 
Which one looks like the most 
unfair
unfair
 distribution?
 distribution?
Each row totals 45
 
Why is Set B most unfair?
Why is Set B most unfair?
Because there is a lot more
Because there is a lot more
Variability
Variability
in the Distribution of the sweets
in the Distribution of the sweets
With your unifix cubes find the Mean, the
With your unifix cubes find the Mean, the
Moves and the Median of Set B.
Moves and the Median of Set B.
Mean/Fair Share = 5
Moves to make fair = 20
Median = 1
How do we think the number of
How do we think the number of
moves might be affected by the
moves might be affected by the
variability in a Distribution?
variability in a Distribution?
Discuss
Discuss
 
“The more variability in a distribution,
 the more moves it takes to make it fair”
Find the Median, Moves & Mean for C, D, E, F
Find the Median, Moves & Mean for C, D, E, F
Each row totals 45
Answers
Answers
Each row totals 45
Each row totals 45
Answers
Answers
Each row totals 45
Answers
Answers
Each row totals 45
Answers
Answers
Do the mean and median always
Do the mean and median always
have to be the same in a
have to be the same in a
Distribution?
Distribution?
Discuss
Discuss
Looking at our Distributions…..
Looking at our Distributions…..
See how the
spread looks
when the
sweets are
represented on
a Dot Plot
A more sophisticated way of
measuring variability or spread
is
Standard Deviation
Deviations from the Mean
Deviations from the Mean
Standard
Standard
 
 
Deviation
Deviation
 
 
= 2.049
 
Use your calculator to calculate the
standard deviation of the various sets
given in the table.
Standard Deviation using Calculator
Standard Deviation using Calculator
Unfair Allocations
Unfair Allocations
Each row totals 45
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Explore the concept of fair distribution through sweets, assessing mean, median, and variability. Engage in activities to make distributions fair by moving items and determining the most equitable distribution among different scenarios. Analyze various distributions of sweets among students and identify the fairest arrangement based on median, moves needed, and mean values.


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  1. Learning Outcomes Understanding the Mean of a Distribution through Fair Share. Engaging with Variability in a Distribution. Measuring Variability through counting the amount of moves needed to make a Distribution fair. Introducing Standard Deviation as a more sophisticated way of measuring Variability.

  2. Key Words Distribution Fair Unfair Mean Variability Spread of a Distribution

  3. The following represents a distribution of 45 sweets shared among 9 students. Is this a fair distribution of the sweets?

  4. Lets make it fair. We need to move around some sweets. How many times will we need to move a sweet to make it fair?

  5. How many moves? 6 moves How many sweets are in a Fair Share? 5 sweets We say The Mean of the distribution is 5 3 moves 1 move 2 moves

  6. Whats the Median of the Distribution?

  7. Whats the Median of the Distribution? 5

  8. Recap: In the below Distribution of Sweets A Fair Share/Mean = 5 No of Moves to make it fair = 6 Median = 5

  9. Here are 6 more Distributions of the 45 sweets 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 6 5 5 4 5 5 6 5 4 A 1 10 10 1 1 10 1 10 1 B 2 4 8 3 4 6 6 7 5 C 4 4 7 4 4 5 6 7 4 D 1 4 8 4 4 6 6 8 4 E 8 1 7 7 4 1 3 7 7 F Each row totals 45

  10. Which one looks like the most fair distribution? 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 6 5 5 4 5 5 6 5 4 A 1 10 10 1 1 10 1 10 1 B 2 4 8 3 4 6 6 7 5 C 4 4 7 4 4 5 6 7 4 D 1 4 8 4 4 6 6 8 4 E 8 1 7 7 4 1 3 7 7 F Each row totals 45

  11. Set A With your unifix cubes find The Mean/Fair Share of the Distribution. Find how many Moves it takes to make set A fair. Find the Median of the Distribution.

  12. Set A Set B Set C Set D Mean/Fair Share = 5 Moves to make fair = 2 Median = 5 Set E Set F

  13. Which one looks like the most unfair distribution? 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 6 5 5 4 5 5 6 5 4 A 1 10 10 1 1 10 1 10 1 B 2 4 8 3 4 6 6 7 5 C 4 4 7 4 4 5 6 7 4 D 1 4 8 4 4 6 6 8 4 E 8 1 7 7 4 1 3 7 7 F Each row totals 45

  14. Set B Why is Set B most unfair? Because there is a lot more Variability in the Distribution of the sweets

  15. Set B With your unifix cubes find the Mean, the Moves and the Median of Set B. I wonder will the answers be different because there is a lot more Variability in the Spread of Set B ????

  16. Set A Set B Set C Set D Mean/Fair Share = 5 Moves to make fair = 20 Median = 1 Set E Set F

  17. How do we think the number of moves might be affected by the variability in a Distribution? Discuss The more variability in a distribution, the more moves it takes to make it fair

  18. Find the Median, Moves & Mean for C, D, E, F 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 6 5 5 4 5 5 6 5 4 A 1 10 10 1 1 10 1 10 1 B 2 4 8 3 4 6 6 7 5 C 4 4 7 4 4 5 6 7 4 D 1 4 8 4 4 6 6 8 4 E 8 1 7 7 4 1 3 7 7 F Each row totals 45

  19. Answers 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 1 6 5 5 4 5 5 6 5 4 A 6 1 10 10 1 1 10 1 10 1 B 3 2 4 8 3 4 6 6 7 5 C 2 4 4 7 4 4 5 6 7 4 D 4 1 4 8 4 4 6 6 8 4 E 5 8 1 7 7 4 1 3 7 7 F Each row totals 45

  20. Answers 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 1 5 6 5 5 4 5 5 6 5 4 A 6 1 1 10 10 1 1 10 1 10 1 B 3 5 2 4 8 3 4 6 6 7 5 C 2 4 4 4 7 4 4 5 6 7 4 D 4 4 1 4 8 4 4 6 6 8 4 E 5 7 8 1 7 7 4 1 3 7 7 F Each row totals 45

  21. Answers 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 2 1 5 6 5 5 4 5 5 6 5 4 A 20 6 1 1 10 10 1 1 10 1 10 1 B 7 3 5 2 4 8 3 4 6 6 7 5 C 5 2 4 4 4 7 4 4 5 6 7 4 D 8 4 4 1 4 8 4 4 6 6 8 4 E 11 5 7 8 1 7 7 4 1 3 7 7 F Each row totals 45

  22. Answers 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean 2 1 5 5 6 5 5 4 5 5 6 5 4 A 20 6 1 5 1 10 10 1 1 10 1 10 1 B 7 3 5 5 2 4 8 3 4 6 6 7 5 C 5 2 4 5 4 4 7 4 4 5 6 7 4 D 8 4 4 5 1 4 8 4 4 6 6 8 4 E 11 5 7 5 8 1 7 7 4 1 3 7 7 F Each row totals 45

  23. Do the mean and median always have to be the same in a Distribution? Discuss

  24. Looking at our Distributions.. The number of moves gives us a Measure of the Variability in the Spread of the Distribution

  25. Set A 2 moves Set B Set C See how the spread looks when the sweets are represented on a Dot Plot Set D Set E Set F

  26. Set A 20 moves Set B Set C Set D Set E Set F

  27. 7 Set A moves Set B Set C Set D Set E Set F

  28. Set A 5 moves Set B Set C Set D Set E Set F

  29. Set A 8 moves Set B Set C Set D Set E Set F

  30. Set A 11 moves Set B Set C Set D Set E Set F

  31. A more sophisticated way of measuring variability or spread is Standard Deviation

  32. Deviations from the Mean

  33. Standard Deviation = 2 ( x ) n = 2.049

  34. Standard Deviation using Calculator Use your calculator to calculate the standard deviation of the various sets given in the table.

  35. Unfair Allocations 1 2 3 4 5 6 7 8 9 Ranking Median Moves Mean S.D. 2 1 5 5 0.67 6 5 5 4 5 5 6 5 4 A 20 6 1 5 4.47 1 10 10 1 1 10 1 10 1 B 7 3 5 5 1.83 2 4 8 3 4 6 6 7 5 C 5 2 4 5 1.25 4 4 7 4 4 5 6 7 4 D 8 4 4 5 2.11 1 4 8 4 4 6 6 8 4 E 11 5 7 5 2.62 8 1 7 7 4 1 3 7 7 F Each row totals 45

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